That is good that the drawer allows for access to the ball valves.
What do you mean that the gate valve is tuned? If you closed this gate valve entirely, the drain rate would drop significantly forcing the water level in the overflow to rise until it flows over the secondary and/or emergency drain enough to match the return rate. Of course, this is an extreme case to show the function of the different drains.
I understand this, but is there a difference in water level between the inner and outer overflows? If the outer overflow is drained too fast in relation to the return rate, the inner overflow will never fill very full because the bulkheads between the two never fill up.
I have a ghost overflow with three drains in the same general configuration as you have and can fine-tune the amount of sound by limiting flow in the primary drain and maintaining a high-enough flow rate. Your return(s) should flow at a high enough rate to allow for the outer overflow to fill to the level of the bottom of the U on the secondary drain. This would allow for the internal overflow to also fill higher to limit the waterfall height to also limit sound.
Does the internal overflow weir have straight or slanted notches? My overflow made by Modular Marine has weir slots with a slope on the inside to allow water to flow down the inside rather than leap off the weir and crash down on the water in the interior overflow.
Hopefully this picture helps explain what I mean: Left side is straight-cut weir slots that allow for gap behind flow-over into interior overflow, Right side is bevel-cut weir slots that allow water to flow smoothly into interior overflow with much less chance for turbulence and open drop onto water surface.